Abstract

In this thesis we look at the structure of racks. Chapter two looks at congruenceson racks. We examine operator group equivalence and associatedgroup equivalence in detail. We show that the fundamental quandle of aknot in S3 embeds into the knot group if and only if the knot is prime. Inchapter three we look at conditions on the associated group and the operatorgroup of a rack. We prove that G is the associated group of a rack only if theassociated group of Conj (G) is isomorphic to G x N, where N is abelian.We also show that any group can be the operator group of a rack. Chapterfour looks at expanding and extending racks. We derive necessary and sufficientconditions for rotation blocks to form a rack when used to expand arack. We also show that any rack, R, can be extended to a complete rackwhich has the same operator group as R. The work in chapter five is closelyconnected to the work of Joyce in [ J ]. We define racks which can be used torepresent any rack. In chapter six we show that the lattice of congruences ona transitive rack is isomorphic to a sublattice of the lattice of subgroups ofthe associated group. We generalize this result to non-transitive racks. Thelast chapter looks at the fundamental rack of a knot in S3.